There are really two cases to consider: a=b and a<>b.
If say a<b then let e=(b-a)/4.
Then there is a positive integer K such that if n>=K then x_n is in (a-e,a+e) and y_n is in (b-e,b+e). In these cases it is easy to see that U_n=y_n and V_n=x_n.